Whisk the warm milk, sugar, and yeast together in a bowl and let it sit for about five minutes. You're looking for a foamy, bubbly top, which tells you the yeast is alive and ready to work. If nothing happens after ten minutes, your milk was likely too hot or too cold, so start over.
Add the yogurt, melted butter, eggs, vanilla, flour, and salt to the yeast mixture and mix on low with a dough hook for one minute. Let it rest for five minutes so the flour can soak up the liquid, then knead on medium for about six minutes until the dough is smooth and only slightly sticky.
Place the dough in a lightly greased bowl, cover it, and set it somewhere warm for about an hour to ninety minutes. It should double in size before you move on. Rushing this step is the easiest way to end up with dense bars later.
While the dough rises, simmer the blueberries, sugar, and lemon juice in a saucepan over medium heat, stirring occasionally. Once the berries break down, stir in the cornstarch mixed with a splash of water and cook until thickened. Cool it completely before filling the bars.
Roll the risen dough into a large rectangle about half an inch thick, then cut it into even three-inch squares. Place each square on its own parchment paper square so it's easy to lift later without deflating it.
Cover the cut bars loosely and let them proof for thirty to forty-five minutes until visibly puffy. This second rise is what gives the bars their soft, cloud-like interior once fried.
Heat your oil to between 350F and 365F, then lower two or three bars at a time into the pot using their parchment squares. Fry for about a minute and a half per side until deeply golden, then transfer to a wire rack to cool slightly.
Whisk together powdered sugar, melted butter, vanilla, and a little warm milk until smooth and pourable. If it thickens while you work, a quick ten seconds in the microwave brings it right back.
Once the bars are just warm to the touch, pipe the cooled blueberry compote into the center of each one until it slightly puffs. Dip the tops in glaze, then finish with whipped buttercream and shortbread crumbles if you're using them.